Sloshing of layered fluid with a free surface as a Hamiltonian system

The aim of this paper is the investigation of a layered sloshing fluid system using both a new Hamiltonian mathematical model and new laboratory experiments. The mathematical model is defined for a cylindrical tank with an arbitrary shape and subjected to an arbitrary rigid motion. The model consists of a pure evolution system of partial differential first order equations in the canonical four unknowns: water elevation at the upper free surface and at the separation surface and the gap in momentum potential density computed at each fluid surface. The system of equations is obtained by avoiding the construction of the Hamiltonian and its variational derivatives. An important advantage of this formulation, with respect to the Lagrangian formulation, is that the nonevolution constraint, which imposes for each fluid the same velocity component along the normal direction of the separation surface, is fulfilled by the model itself. The model implementation needs to define the so-called Neumann-Dirichlet operators, which are computed by an efficient algorithm, for any instantaneous configuration of the two fluid domains. A numerical integration of the model is performed by a suitable Galerkin projection of the evolution equations. New laboratory experiments, simulating the sloshing of a layered fluid system, inside a tank with a squared cross section, were performed. The experiments, with a two-dimensional sloshing, were carried out by varying the forcing frequency, the oscillation amplitude and the ratio of the two fluids' depths. In some experiments, a traveling wave, with a shape similar to a moving hydraulic jump, was observed at the separation surface. Measurements of the space-time evolution of both the free and the separation surfaces were performed and compared with the model's predictions. A good agreement between the model predictions and laboratory measurements is found, even for strong nonlinear cases such as the traveling wave.